1. **Problem statement:** Given a circle with center $C$ and radius $41$ mm, a tangent line $QP$ at point $Q$ on the circle with length $QP=9$ mm, find the length of chord $CS$. 2. **Key concepts:** - The radius $CQ$ is perpendicular to the tangent $QP$ at point $Q$. - The tangent-secant theorem states that the square of the tangent segment length equals the product of the secant segment lengths. 3. **Setup:** - Since $CQ$ is radius, $CQ=41$ mm. - $QP=9$ mm. - We want to find chord $CS$. 4. **Using the tangent-secant theorem:** - Let $CS = x$ mm. - The tangent-secant theorem: $$QP^2 = CS \times CR$$ where $CR$ is the other segment of the secant line passing through $C$. 5. **Since $C$ is center and $CS$ is chord, $CR$ is radius $41$ mm.** - So, $$9^2 = x \times 41$$ 6. **Solve for $x$:** $$81 = 41x$$ $$x = \frac{81}{41}$$ 7. **Final answer:** $$CS = \frac{81}{41} \approx 1.9756$$ mm Thus, the length of chord $CS$ is approximately 1.9756 mm.